Margo on Meantone

[A440A@AOL.COM]

 Greetings, 
    I have received several questions concerning Meantone history, and think 
that  this recent posting from    Margo Schulter" <MSCHULTER@VALUE.NET> goes 
far beyond what I could offer. 

enjoy or delete! 
Ed Foote 

Subject: Re: quarter-comma meantone on keyboard


Hello, there, Alita Morrison, and everyone.

Please let me begin by saying that like many other tunings, 1/4-comma
meantone can be precisely defined in mathematical terms, but will vary
in "real world" realizations, whether on a 16th-century harpsichord
tuned by ear or on a modern synthesizer with limited pitch resolution.
 
(Around 1600 in Naples, a 19-note instrument with split keys was
rather popular, the Scala scale archive includes a file for this
tuning also, meanquar_19.scl.)

In 1571, Gioseffo Zarlino published what may be the first
comprehensive mathematical description of this tuning, which Aaron
(1523) may have defined more informally, and which Vicentino (1555)
likely used for his archicembalo with 36 notes per octave (in his
first tuning, 31 of them dividing the octave into nearly equal parts
of a circulating system, and the other five being used to provide pure
fifths for a few common notes). Interestingly, the great Spanish
theorist Salinas (1577) reports that he was considered the "inventor"
of this tuning during his stay in Rome about 40 years earlier.

The special quality of 1/4-comma meantone is that it has pure major
thirds at 5:4. Whether many 16th-century tuners focused especially on
these thirds is an open question, and Aaron's instructions of 1523
present an interesting case.

He says to start by tuning the octave C-C and the third C-E "as
sonorous and just as possible," with "the greatest possible unity."
Then the major third C-E is used as the basis for a chain of four
fifths, each of which should be a bit narrower than pure by the same
amount: C-G-D-A-E. Then the rest of the tuning is completed by moving
in fifths in either direction.

Zarlino more precisely defines that each fifth should be narrow by an
amount equal to 1/4 of the syntonic comma (81:80, ~21.51 cents), the
amount by which four pure fifths minus two octaves would exceed a pure
5:4 third. Using the logarithmic measure of cents, logarithms and
their application to music developing in the 17th century, we can say
that such a fifth is approximately 5.38 cents narrower than pure, with
a size of about 696.5784 cents.
 
By the way, I've played 16th-century European music and improvised in
related styles on a synthesizer with two independently tuned keyboards 
to obtain between 13 and 24 notes per octave in 1/4-comma meantone.
One arrangement popular in Naples around 1600 was a 19-note instrument
(Gb-A#), and Scala has a number of files in 1/4-comma meantone for
differing numbers of notes per octave.

With 24 notes on two keyboards, you get a subset of Vicentino's
archicembalo with its 31-note meantone cycle. The keyboards differ by
Vicentino's "diesis" or fifthtone, often 128:125 or ~41.06 cents in
1/4-comma meantone (the distance between G# on the lower keyboard and
Ab on the higher one, for example).

Note that while Vicentino considers his tuning to divide the
whole-tone into five equal parts, he also appears to consider the
major thirds "perfect" or pure -- two characteristics which it was
shown by the later 17th century actually define two different
tunings, 31-tone equal temperament (31-tET) and 1/4-comma meantone.
In practice, variations in tuning by ear might be greater than the
theoretical difference between these two tunings, both of which
circulate nicely in 31 notes.

Also, while Vicentino himself noted that every interval was available
in his 31-note cycle from every note on the instrument, circularity
wasn't an important factor in even experimental 16th-century keyboard
music, although a few pieces (e.g. a puzzle piece for voices by
Willaert) may suggest such a closed system. In 1618, Fabio Colonna did
publish an "example of circulation" going through a cycle of 31 fifths
on his superharpsichord, somewhat similar to Vicentino's but with a
different keyboard layout, the _Sambuca Lincea_.
 

Most respectfully,

Margo Schulter
mschulter@value.net